Question: Solve for $x$ and $y$ using elimination. $\begin{align*}6x-7y &= -9 \\ -9x+4y &= -6\end{align*}$
Answer: We can eliminate $x$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $3$ and the bottom equation by $2$ $\begin{align*}18x-21y &= -27\\ -18x+8y &= -12\end{align*}$ Add the top and bottom equations. $-13y = -39$ Divide both sides by $-13$ and reduce as necessary. $y = 3$ Substitute $3$ for $y$ in the top equation. $6x-7( 3) = -9$ $6x-21 = -9$ $6x = 12$ $x = 2$ The solution is $\enspace x = 2, \enspace y = 3$.